The minima of the geodesic length functions of uniform filling curves

Abstract

There is a natural link between (multi-)curves that fill up a closed oriented surface and dessins d'enfants. We use this approach to exhibit explicitly the minima of the geodesic length function of filling curves that admit a self-transverse homotopy equivalent representative such that all self-intersection points, as well as all faces of the complement, have the same multiplicity. We show that these minima are attained at the Grothendieck–Belyi surfaces determined by the natural dessin d'enfants associated with these filling curves. In particular, they are all Riemann surfaces defined over number fields.